Go through these in order and stop at the first one that applies. If you are reading this for someone else, answer for them.
A cold, pale or blue hand, loss of feeling, or rapidly increasing swelling after an injury needs emergency assessment.
Get urgent medical help after a major injury if the elbow looks deformed, you cannot move it, or there was a snap followed by sudden weakness or extensive bruising.
A hot, red, very swollen elbow with fever also requires prompt medical care because infection or another inflammatory condition may be present.
Arrange a routine appointment if elbow pain has not improved after several weeks of sensible activity changes, or if it is getting worse.
Assessment is also appropriate if pain regularly disturbs sleep, limits work or daily activities, or the diagnosis is uncertain.
The article also says
Seek earlier medical advice if you have numbness, tingling, marked weakness, repeated dropping of objects, significant swelling or difficulty moving the elbow.
These are not typical features of uncomplicated tennis elbow.
Every sign above is quoted from the sofidex article on Tennis elbow. If you are not sure, err upward — it is always reasonable to be checked.